Abstract

We will study the spectrum for the biharmonic operator involving the laplacian and the gradient of the laplacian with weight, which we call third-order spectrum. We will show that the strict monotonicity of the eigenvalues of the operator , where , holds if some unique continuation property is satisfied by the corresponding eigenfunctions.

1. Introduction

We are concerned here with the eigenvalue problem:
where is a bounded domain in , , denotes the biharmonic operator defined by , and .

Based on the works of Anane et al. [1, 2], we will determine the spectrum of (1.1), which we call third-order spectrum for the biharmonic operator. This spectrum is defined to be the set of couples such that the problem
has a nontrivial solution . This spectrum, which is denoted by , is an infinite sequence of eigensurfaces , see Section 3. When , the zero-order spectrum is defined to be the set of eigenvalues such that the problem
has a nontrivial solution . In this case the spectrum is denoted by . The eigenvalue problem (1.3), which is studied by Courant and Hilbert [3], admits an infinite sequence of real eigenvalues satisfying
where denotes the class of -dimensional subspaces of .

Definition 1.1. We say that solutions of problem (1.1) satisfy the unique continuation property (U.C.P), if the unique solution which vanishes on a set of positive measure in is .

In the literature there exist several works on unique continuation. We refer to the works of Jerison and Kenig [4] and Garofalo and Lin [5], among others. The unique continuation property as defined above differs from the usual notions of unique continuation, see [6] for more details.

Definition 1.2. We say that is strict monotone with respect to the weight if , for all .

Here we use the notation to mean inequality almost everywhere together with strict inequality on a set of positive measure.

Since the pioneer works of Carleman [7] in 1939 on the unique continuation, this notion has been the interest of many researchers in partial differential equations, see for instance [4, 5, 8]. In 1992, de Figueiredo and Gossez [6] proved that strict monotonicity holds if and only if some unique continuation property is satisfied by the corresponding eigenfunction of a uniformly elliptic operator of the second order. In 1993, Gossez and Loulit [8] have proved the unique continuation property in the linear case of the laplacian operator. The unique continuation property of the biharmonic operator was proved recently by Cuccu and Porru [9]. Our purpose in the fourth section is to show that strict monotonicity of eigensurfaces for problem (1.1) holds if some unique continuation property is satisfied by the corresponding eigenfunctions.

2. Preliminaries

Let be a finite dimensional separable Hilbert space. We denote by and the inner product and the norm of the space , respectively. Let be a compact operator.

Lemma 2.1. All nonzero eigenvalues of the operator are obtained by the following characterizations:
where denotes the class of -dimensional subspaces of .

Moreover, zero is the only accumulation point of the set of all eigenvalues of . Here, the eigenvalues are repeated with its order of multiplicity, and the eigenfunctions are mutually orthogonal [10].

3. Third-Order Spectrum of the Biharmonic Operator

We define the third-order eigenvalue problem of the biharmonic operator as follows:

If is a solution of (3.1) then is called third-order eigenvalue and is said to be the associated eigenfunction.

Lemma 3.1. Problem (3.1) is equivalent to the following problem:
where .

Proof. For any , we have
Hence, problem (3.1) is equivalent to problem (3.2)

Remark 3.2. Let ; we denote by the normal derivative defined by where and .

Definition 3.3. A weak solution of (3.2) is a function in witch satisfies, for and for all ,

Definition 3.4. For , we say that is a classical solution of problem (3.1) if .

Proposition 3.5. If is a weak solution of (3.2) and , then is a classical solution of (3.2).

Proof. Let be a weak solution of (3.2), then we have
Using the Green formula, we obtain
Then we have
Thus, the prove is complete.

Theorem 3.6. Let , then we have (a), where the function : is defined by
where denotes the class of -dimensional subspaces of and is the graph of .(b). (c)For all .

Proof. Let , then is a solution of (3.1) if and only if is a solution of problem (3.2). We prove that the map
defines a scalar product on equivalent to the usual scalar product .The map is a continuous symmetric bilinear form. Since satisfies the condition of the uniform ellipticity, then we have
where . Therefore, the bilinear form is coercive. On the other hand, the operator
is well defined, linear, symmetric, and compact on . Then, problem (3.2) can be written as
Note that is not an eigenvalue of (3.2). It follows that is an eigenvalue of (1.1) if and only if is eigenvalue of the operator . By Lemma 2.1, the eigenvalues are given by the characterizations
In addition, we have
and , then relation (3.8) is satisfied. Since , then we have for all . As zero is the only accumulation point of the sequence , it follows that when . Therefore, the proof is completed.

4. Strict Monotonicity and Unique Continuation

In this section, we will show that strict monotonicity of eigensurfaces for problem (3.1) holds if some unique continuation property is satisfied by the corresponding eigenfunctions.

Theorem 4.1. Let and be two weights with and . If the eigenfunctions associated to satisfy the (U.C.P) then .

Theorem 4.2. Let be a weight and . If the eigenfunctions associated to do not satisfy the (U.C.P) then there exists a weight with , such that, for some with , one has .

As a consequence of Theorems 4.1 and 4.2 we have the following result.

Corollary 4.3. Let and . If , then the only solution of the problem
is .

Proof of Theorem 4.1. Let ; we define the space
spanned by the eigenfunctions associated to with
We have
Let , with . We have either achieves the infimum in (4.4) or not. In the case is an eigenfunction associated to , then by the (U.C.P) and since a.e. , we have
Thus, . In the other case, we have
Thus, in both cases we have
It follows that
This yields the desired inequality . Hence, we have .

Proof of Theorem 4.2. Denote by an eigenfunction associated to which vanishes on a set of positive measure. Take such that and define
where is chosen such that , which is possible by the continuous dependence of the eigenvalues with respect to the weight. We have
which shows that is an eigenvalue for the weight , that is, for some . Let us choose the largest such that this equality holds. It follows from that . Moreover, the monotone dependence, , implies . Then we conclude that . Hence, we have .

Proof of Corollary 4.3. Suppose that has nontrivial solution, that is, . From the inequality and the strict monotonicity, we deduce
Since
we deduce that
which is a contradiction. Hence, the proof is complete.

Acknowledgment

The authors would like to thank the anonymous referees for their constructive suggestions.